On Achieving Capacity on the Wire Tap channel using LDPC Codes

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1 On Ahievin Capaity on the Wire Tap hannel usin LDPC Codes Andrew Thanaraj Indian Institute of Tehnoloy Madras India teven W. MLauhlin GTL-CNR Lab Metz Frane ouvik Dihidar Georia Institute of Tehnoloy Atlanta Jean-Mar Merolla GTL-CNR Lab Metz Frane A. Robert Calderbank Prineton University Abstrat We investiate the use of apaity and near-apaity ahievin LPDC odes on the wire tap hannel where the dual onditions of reliable ommuniations and seurity are reuired. We show that ood odes for onventional hannels (like BC and BEC) also have interestin and useful seurity properties. In this paper we show the onnetion between the deodin threshold of the ode and its seurity aainst eavesdroppin. We also ive pratial ode onstrutions for some speial ases of the wire tap hannel and show that seurity (in the hannon sense) is a funtion of the deodin threshold. ome of these onstrutions ahieve the serey apaity as defined by Wyner. These odes provide seure ommuniations without onventional key distribution and provide a physial-layer approah for either seure ommuniations or key distribution. Alie Alie s Enoder Fi.. X D D 2 Eve The wiretap hannel system Bob Y I. INTRODUCTION The notion of ommuniation with perfet seurity was defined in information-theoreti terms by hannon [. uppose a -bit messae is to be transmitted seurely from Alie to Bob aross a publi hannel. Perfet seurity is said to be ahieved if the enodin of into a transmitted word is suh that the mutual information.from this definition hannon onluded that Alie and Bob should neessarily share bits of key for ahievin perfet seurity. An alternative notion of ommuniation with perfet seurity was introdued by Wyner [2 for the more eneral wire tap hannel. In a eneral wire tap hannel system (Fi. ) and are disrete memoryless hannels (DMCs). The two DMCs have the same input alphabet but may have different output alphabets. and of a wire tap hannel system are alled the main hannel and the wire tap hannel respetively. Wyner s notion of serey apaity is the maximum possible rate of information transmission between Alie and Bob that still keeps Eve totally inorant. If the main hannel is less noisy than the wire tap hannel [4 then the serey apaity is " $ ' * ' / where the maximum is over all possible hannel input distributions 5 6 of '. Wyner showed that if is a deraded version of ( is onatenated with another DMC) then serey apaity is positive. Csiszár et al. [4 showed that the serey apaity is positive for the ases when is less noisy than. () Maurer [5 [6 eneralized this even further and was able to onlude several very powerful results for a eneral and. In this paper we investiate the use of LDPC-based odes for perfetly seure information exhane in some speial ases of wire tap hannel systems with a binary erasure hannel (BEC) or binary symmetri hannel (BC) for the wire tapper s hannel. There are several results in this paper: ) We make the onnetion between the threshold of LDPC odes (under messae-passin deodin) and perfet seurity. 2) For the ase of a noiseless main hannel and BEC wire tap hannel we onstrut odes based on LDPC odes that ahieve perfet seurity and have rates approahin the serey apaity. ) For the ase of the BEC main and BEC wire tap hannel we provide ode onstrutions and ive onditions for ahievin perfet seurity on the wire tap hannel and zero probability of error on the main hannel. Perfet seurity in this ase is ahieved only when apaityahievin erasure-orretin odes are used and the ode rate is below serey apaity. 4) For the ase of a noiseless main hannel and a binary symmetri hannel (BC) as the wiretapper s hannel we provide a odin solution usin odes that have ood error-detetin apability. Perfet seurity is ahieved asymptotially but at rates tendin to zero. The losest previous work to this is in [ where ertain

2 7 t t o aspets suh as the existene of LDPC odes for odin over wire tap hannels have been studied in the ontext of key eneration from orrelated soure outputs; however the ontext and methods in our work are different than [. The odin problem for Alie in the wire tap hannel system involves addin redundany for enablin Bob to orret errors (aross the main hannel) and addin randomness for keepin Eve inorant (aross the wire tap hannel). Let us assume that Alie needs to transmit one out of eually likely messaes i.e. a messae denoted (denoted as in Fi. ) is suh that and Prob. Alie uses odes with and blok-lenth. A messae is enoded into a transmitted word as follows: is hosen uniformly at random from the ode. The transmitted word in eneral belons to the overall ode. The rate of information transmission from Alie to Bob (in terms of bits per hannel use) in suh a settin is iven by! " $. The reeiver on the main hannel (Bob) deodes a reeived word & with respet to the overall ode into a deoded messae ' (say by Maximum-Likelihood deodin). The eavesdropper on the wire tap hannel is assumed to have unlimited power to proess the reeived word (. The objetive of Alie and Bob in a wire tap hannel system an now be iven a preise definition. Let ) ) * and + be random variables denotin Alie s messae Bob s deoded messae and Eve s reeived word respetively. Let - represent the entropy of a random variable -. Then the objetive is to ahieve the followin: ) + 5 / 2 ) 4 ) ) * 7! " $ (2) () The onstraint () is referred to as the seurity onstraint while (2) is alled the reliability onstraint. If an enoder (as in Fi. ) with?! " $ satisfies the seurity and reliability onstraints for a iven wire tap hannel system then suh an enoder is said to ahieve a serey rate?. In [2 we have shown that if eah is hosen to be a apaity-ahievin ode on and is hosen to be a apaity-ahievin ode on then we an ahieve the serey apaity of any wire-tap system (Fi. ) where both and are DMCs. We now ive a sketh of the proof in the next setion. II. PROOF OF EXITENCE OF CODE AND CODING METHOD A suffiient ondition for perfet seurity on the wire tap hannel is : Eah should approah apaity over the wire tap hannel (similar to the speial ase onsidered by Wyner in [2). We present the riterion in the followin theorem without proof. Theorem : If the odes ahieve apaity over the wire tap hannel then Prob ) + ( 5 / 2 ). 5 I M N O??? U U 5 I $ W X? $ W \? We now fous on the probability of error on the main hannel. Let be the probability of erro averaed over an ensemble of odes and all possible transmitted messaes. Let ;. Note that the serey rate of a ode from the ensemble is.wenowivethe followin theorem without proof. Theorem 2: The averae probability of error is bounded by where W X? and W \? are the random odin exponents for the wire tap hannel and the main hannel respetively. We know that W \? for? ^ \ where \ is the hannel apaity of the wire tap hannel. Hene Theorem 2 says that there exists a ode in a suitable ensemble suh that the seurity onstraint an be satisfied (eah an approah apaity on the wire tap hannel) with arbitrary auray by inreasin the blok-lenth; at the same time the same ode an satisfy the reliability onstraint with arbitrary auray provided the rate? is suh that W X?. Hene the maximum serey rate ahievable by a ode from the ensemble is ` a \ where` is the distribution on that maximizes the random odin exponent W \?. Thus if both the main hannel and wire tap hannel are symmetri serey apaity is ahievable. We now study the desin and use of linear odes over a wire tap hannel system. To transmit -bit messaes we first e e selet a linear binary ode suh that.out of the h i osets of we hoose k osets and let eah messae orrespond to a hosen oset. The seletion of the osets is done in a linear fashion. uppose l is a enerator matrix for with rows m m andm i. We selet linearly independent vetors n n andn k from. Let l be the matrix with rows as n n andn k.the oset orrespondin to a -bit messae s $ t t t k is determined as follows: s 7 t n t n k n k (4) Thouh the above orrespondene is deterministi the enodin proedure has a random omponent in the seletion of the transmitted word. A -bit messae s is enoded into a -bit word randomly seleted from the oset of orrespondin to s. Hene the transmitted word isivenby t n t n k n k u m u m u i m i where x $ u u u i is an uniformly random e -bit vetor. III. ERAURE WIRE TAP YTEM We now onentrate on wire-tap systems in whih (Fi. ) is a binary erasure hannel (BEC). uh systems appear in the uantum key distribution problems. In [2 we onsidered the problem where is a BEC and is noiseless. We showed that dual odes of ood erasure orretin LDPC odes an be used to desin odes for suh systems. We used the followin result from [ Lemma.

3 l? ( 4 $ O P O Theorem (Ozarow Wyner 84): Let an ode have a enerator matrix l $ where is the - th olumn of l. Consider an instane of the eavesdropper s observation ( with unerased positions iven by. ( is seured by iff the matrix has rank. We now ive the followin examples from [2. Example : Let be a BEC with erasure probability. The ensemble of -reular LDPC odes has threshold.let be an adjaeny matrix from the ensemble with lare (say ). is an binary matrix with row weiht and olumn weiht.the ode dual of with enerator matrix an be used for with perfet serey. The information rate between the honest parties in this ase is? ompared to the upper bound of (from ()). Example 2 (Tornado odes): A rate- tornado ode ensemble with threshold " has been reported in [0. $ A parity-hek matrix for a ode from the ensemble will have dimensions.the ode dual of with enerator matrix an be used over an erasure $ wire tap hannel for with perfet serey. The information rate between the honest parties in this ase is ompared to the upper bound of $. A. Erasure main hannel and erasure wire tap hannel In this setion both and are BECs with erasure probabilities \ and X respetively. Our results apply with a small modifiation to systems with DMCs other than the BEC as. Aordin to () the serey apaity of this system is \ X whih is positive whenever \ X. We first pik an LDPC ode of lenth from an ensemble of odes havin asymptoti erasure threshold \. That means as 7 ' reovers all the erasures on an erasure hannel with erasure probability up to at least \ usin the standard iterative erasure deodin alorithm. Let have rate / andlet be the parity hek matrix of the ode. Next we pik / independent vetors from the dual spae of where/ ^ /.Let be the matrix formed by these vetors as rows. has dimensions /.Let be the rest of the independent vetors in the dual spae of. As we will see shortly we must have \ / in order to uarantee some euivoation for Eve. Let be the parity hek matrix of a ode.wewant to have asymptoti erasure threshold X.Wethenhave and / X / \ We now disuss the enodin proedure. Alie first takes a / / -bit lon messae vetor and forms a / - bit lon vetor by addin / s on top of. he now hooses an at random from the solution set of the euation shown in Fiure 2 and transmits it. (5) (6) n(-r 2 ) n(r 2 -r ) Fi. 2. H 2 H 2 X = The enodin proedure We illustrate this enodin proedure in Fiure. Note that the number of solutions to the euation h is h. P ". P. However for some partiular hoie of say / the number of solutions to the euation shown in Fi. 2 is h h. O ". O. Obviously the same annot be a solution for two different values of. This explains the splittin of the solution set spae of the euation into 4 P. P h. O " 4 disjoint subsets eah orrespondin to a different value of. Hene the rate of our ode is / /. The interestin point to observe in Fiure is that we are not usin the whole spae of. solution set of Fi. 2 for = solution set of H 2 X= Fi.. solution set of H 2 X=0 The enodin spae 0 solution set of Fi. 2 for = 2 solution set of H 2 X= 2 B. Euivoation aross the wire tap hannel In this setion we alulate the euivoation for Eve. ine Eve s hannel is a BEC with erasure probability \ with probability tendin to Eve will have \ erasures as 7 '.Ifwehave \ / usin Evemust

4 ' have at least h h. P " " solutions for all of whih are eually likely. All these solutions will differ from eah other in the erased positions. ine \ is the erasure threshold of the ode havin as the parity-hek matrix any submatrix formed usin \ olumns of will have full olumn rank. Thus every solution of will ive a different value of all of whih are eually likely. The euivoation for Eve is then \ /.If is the parity-hek matrix of a apaity-ahievin ode on an erasure hannel with erasure probability \ / / and the messae will be ompletely seure from Eve. C. Probability of error on the main hannel When Bob reeives a vetor he first deodes it by usin the standard iterative erasure deodin tehniue for LDPC odes on the Tanner raph of the ode. Let the erasure probability of the main hannel be at most X. Then as 7 with probability tendin to he will be able to reover the transmitted word. Bob then an find out the produt whih is his estimate of the messae. Example : Let be a -reular LDPC ode with blok-lenth. has rate /. The ode is hosen to be another LDPC ode with all variable nodes havin deree and all hek nodes havin deree. has rate /. The LDPC ode has an erasure threshold. The ode has an erasure threshold. Thus the serey rate is / / andan euivoation of / is uaranteed aross the wiretap hannel havin erasure probability reater than. Bob an deode the messae with asymptotially zero probability of error on the main hannel havin erasure probability at most. D. Remarks Like in the ase of noiseless main hannel we ould have hosen to be an LDPC ode with erasure threshold \. wil then have to be ontained in whih has erasure threhold X. The matries and will be the parity-hek matries of and respetively. ine the dual of an LDPC ode is likely to have a sinifiantly hih number of low-weiht odewords this reuirement appears to be ontrary to intuition. A very similar ode desin problem arises in the onstrution of uantum error-orretin odes usin sparse raphs [7. After studyin several onstrutions the authors of [7 onlude that suh odes are diffiult to onstrut and are unlikely to have hih thresholds. IV. NOIELE MAIN CHANNEL AND BC WIRETAP CHANNEL In this setion is noiseless and is a binarysymmetri hannel (BC) with error probability in Fi.. We let be an ode and be the entire spae. For an arbitrary -bit messae s the transmitted word s l. ee setion II for the definition of l. ine the osets of over the entire spae of -tuples Eve s reeived vetor / belons to some oset of say l.if denotes the error vetor introdued by the BC in the wire tap hannel we have 5 / 2 / l and s 5 / 2 s l 5 / 2 s l 5 / 2 (7) (8) for some -tuple. We an now state the riterion for seletin the ode to uarantee seurity of the messae : we hoose suh that for any -tuple wehave 5 / 2 () Usin the above ondition in (7)(8) we see that Eve is eually likelytofind+ in any oset of iven any messae s. Assumin all s are eually likely aprioriprob + l is independent of ; hene Prob s + l and perfet seurity is uaranteed. Usin the MaWilliams identities [8 Pae 27 for the linear ode weet h (0) h \ " \ " k where is the number of odewords of weiht in the dual ode.usin and in (0) we et \ " h \ " Usin the MaWilliams identities [8 Pae 7 for the oset weet h " $ where h \ " \ " k () (2) with eual to the number of odewords of weiht in the dual ode orthoonal to and eual to the number of odewords of weiht in the dual ode not orthoonal to.usin and in () we et " $ \ " h \ " () From (2) we see that. We now state the main seurity riterion as a theorem without proof. Theorem 4: If then Prob for all -tuples. (4)

5 ^? 7 We now provide some examples that satisfy the reuirement of (4). Example 4: (inle parity hek odes) The dual of a sinle parity hek ode is the repetition ode with weiht distribution and. Hene for lare. However the serey rate 7 for lare. This is an example that was first used by Wyner in [2 to motivate odin over a wire tap hannel system. Example 5: (Hammin odes) The weiht distribution of the dual of the $ X Hammin ode X is. Hene and $ " $ " for lare. As in the previous example the serey rate tends to zero for lare. We now state the followin theorem without proof that eneralizes the above onstrution method. Theorem 5: Let " be a seuene of odes h k suh that Prob Detetion Error over a BC with error probability and! ^! " $.Let be the number of odewords of weiht in the dual ode ". Then for any -tuple as 7 ' The existene of linear odes with probability of detetion error less than h k is well known [ etion.6. uppose we find a lass of suh error detetin odes suh that! Then for lare the ode " when used as the ode over a wire tap hannel system with a BC (with error probability ) as the wiretapper s hannel provides h perfet N seurity whenever?! " $ or.the maximum possible serey rate that an be ahieved by this onstrution is therefore! " $. Codes suh as Hammin odes and double error-orretin BCH odes are examples of suh error-detetin odes. However most known lass of suh odes have? asymptotially. system with a BEC as the wiretapper s hannel. This result enables the use of odes that an be more easily onstruted. For the ase where both the main hannel and the wiretapper s hannel are BECs we have studied two approahes for ode desin. The optimality and serey apaity of the onstrutions need to be studied and explored. For the ase where the wiretapper s hannel is a BC (with error probability ) and the main hannel is noiseless we have shown that odes with ood error-detetin properties provide seurity. The apaity of this onstrution is! " $ whih is less than the serey apaity. Capaity-approahin odes will probably be raph-based. Use of raph-based odes for the BC wiretapper s hannel is a subjet for future study. REFERENCE [ C. E. hannon Communiation Theory of erey ystems Bell yst. Teh. J. vol. 28 pp Ot. 4. [2 A. D. Wyner The Wire-tap Channel Bell yst. Teh. J. vol. 54 no. 8 pp [ L. H. Ozarow and A. D. Wyner Wire-Tap Channel II Bell yst. Teh. J. vol. 6 pp De. 84. [4 I. Csiszar and J. orner Broadast Channels with Confi dential Messaes IEEE Trans. Inform. Theory vol. IT-24 pp. -48 May 78. [5 U. Maurer eret ey Areement by Publi Disussion from Common Information IEEE Trans. Info. Theory vol. no. pp [6 U. Maurer Unonditionally eure Ey Areement and the Intrinsi Conditional Information IEEE Trans. Info. Theory vol. 45 no. 2 pp [7 D. Makay G. Mithison and P. MFadden parse Graph Codes for Quantum Error-orretion uant-ph/0046 ubmitted to IEEE Transations on Information Theory. [8 F. J. MaWilliams and N. J. loane The Theory of Error-orretin Codes Amsterdam The Netherlands: North-Holland 77. [ J. Muramatsu eret ey Areement from Correlated oure Outputs usin Low Density Parity Chek Matries IEICE Trans. Fundamentals vol. E87-A no. 0 pp [0 P. Oswald and A. hokrollahi Capaity Ahievin eunes for the Erasure Channel IEEE Trans. Info. Theory vol. 48 pp De [. Lin and D. J. Costello Jr. Error-ontrol Codin: Fundamentals and Appliations Enlewood Cliffs NJ: Prentie Hall In. 8. [2 A. Thanaraj. Dihidar A. R. Calderbank. MLauhlin and J. M. Merolla Capaity Ahievin Codes for the Wire Tap Channel with Appliations to Quantum ey Distribution ubmitted to IEEE Transations on Information Theory preprint available at V. CONCLUION When the wiretapper s hannel is a BEC and the main hannel is noiseless we have presented odes that approah serey apaity. To our knowlede these are the first and only suh odes. However we have shown that apaity-ahievin odes are not neessary in this ase. If a ode exhibits a threshold behavior aross a BEC (odes suh as reular LDPC odes) its dual an be used effetively over a wire tap hannel